3.227 \(\int \cosh (x) \cosh (4 x) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x) \]

[Out]

1/6*sinh(3*x)+1/10*sinh(5*x)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4283} \[ \frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Cosh[4*x],x]

[Out]

Sinh[3*x]/6 + Sinh[5*x]/10

Rule 4283

Int[cos[(a_.) + (b_.)*(x_)]*cos[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
+ Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin {align*} \int \cosh (x) \cosh (4 x) \, dx &=\frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {1}{6} \sinh (3 x)+\frac {1}{10} \sinh (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Cosh[4*x],x]

[Out]

Sinh[3*x]/6 + Sinh[5*x]/10

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fricas [B]  time = 0.41, size = 34, normalized size = 2.00 \[ \frac {1}{10} \, \sinh \relax (x)^{5} + \frac {1}{6} \, {\left (6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{3} + \frac {1}{2} \, {\left (\cosh \relax (x)^{4} + \cosh \relax (x)^{2}\right )} \sinh \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x, algorithm="fricas")

[Out]

1/10*sinh(x)^5 + 1/6*(6*cosh(x)^2 + 1)*sinh(x)^3 + 1/2*(cosh(x)^4 + cosh(x)^2)*sinh(x)

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giac [B]  time = 0.11, size = 27, normalized size = 1.59 \[ -\frac {1}{60} \, {\left (5 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{20} \, e^{\left (5 \, x\right )} + \frac {1}{12} \, e^{\left (3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x, algorithm="giac")

[Out]

-1/60*(5*e^(2*x) + 3)*e^(-5*x) + 1/20*e^(5*x) + 1/12*e^(3*x)

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maple [A]  time = 0.23, size = 14, normalized size = 0.82 \[ \frac {\sinh \left (3 x \right )}{6}+\frac {\sinh \left (5 x \right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*cosh(4*x),x)

[Out]

1/6*sinh(3*x)+1/10*sinh(5*x)

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maxima [B]  time = 0.33, size = 27, normalized size = 1.59 \[ \frac {1}{60} \, {\left (5 \, e^{\left (-2 \, x\right )} + 3\right )} e^{\left (5 \, x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} - \frac {1}{20} \, e^{\left (-5 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x, algorithm="maxima")

[Out]

1/60*(5*e^(-2*x) + 3)*e^(5*x) - 1/12*e^(-3*x) - 1/20*e^(-5*x)

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mupad [B]  time = 1.44, size = 15, normalized size = 0.88 \[ \frac {8\,{\mathrm {sinh}\relax (x)}^5}{5}+\frac {8\,{\mathrm {sinh}\relax (x)}^3}{3}+\mathrm {sinh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(4*x)*cosh(x),x)

[Out]

sinh(x) + (8*sinh(x)^3)/3 + (8*sinh(x)^5)/5

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sympy [A]  time = 0.41, size = 20, normalized size = 1.18 \[ - \frac {\sinh {\relax (x )} \cosh {\left (4 x \right )}}{15} + \frac {4 \sinh {\left (4 x \right )} \cosh {\relax (x )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*cosh(4*x),x)

[Out]

-sinh(x)*cosh(4*x)/15 + 4*sinh(4*x)*cosh(x)/15

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